In mathematics, more specifically in ring theory, a Euclidean domain is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of integers. This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements. In particular, the greatest common divisor of any two elements exists and can be written as a linear combination of them. Also every ideal in a Euclidean domain is principal, which implies a suitable generalization of the fundamental theorem of arithmetic: every Euclidean domain is a unique factorization domain.
In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors. For example,
In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c.
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as or
The imaginary unit or unit imaginary number is a solution to the quadratic equation x2 + 1 = 0. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of i in a complex number is 2 + 3i.
In category theory, an epimorphism is a morphism f : X → Y that is right-cancellative in the sense that, for all objects Z and all morphisms g1, g2: Y → Z,
In mathematics, a unique factorization domain (UFD) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain in which every non-zero non-unit element can be written as a product of irreducible elements, uniquely up to order and units.
In number theory, the ideal class group of an algebraic number field K is the quotient group JK /PK where JK is the group of fractional ideals of the ring of integers of K, and PK is its subgroup of principal ideals. The class group is a measure of the extent to which unique factorization fails in the ring of integers of K. The order of the group, which is finite, is called the class number of K.
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. It is a specific example of a quotient, as viewed from the general setting of universal algebra. Starting with a ring R and a two-sided ideal I in R, a new ring, the quotient ring R / I, is constructed, whose elements are the cosets of I in R subject to special + and ⋅ operations.
In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries . In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by William Thurston (1982), and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture.
In topology, the Jordan curve theorem asserts that every Jordan curve divides the plane into an "interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior points. Every continuous path connecting a point of one region to a point of the other intersects with the curve somewhere. While the theorem seems intuitively obvious, it takes some ingenuity to prove it by elementary means. "Although the JCT is one of the best known topological theorems, there are many, even among professional mathematicians, who have never read a proof of it.". More transparent proofs rely on the mathematical machinery of algebraic topology, and these lead to generalizations to higher-dimensional spaces.
In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if Q is a submodule of some other module, then it is already a direct summand of that module; also, given a submodule of a module Y, any module homomorphism from this submodule to Q can be extended to a homomorphism from all of Y to Q. This concept is dual to that of projective modules. Injective modules were introduced in and are discussed in some detail in the textbook.
Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called K-groups. These are groups in the sense of abstract algebra. They contain detailed information about the original object but are notoriously difficult to compute; for example, an important outstanding problem is to compute the K-groups of the integers.
In mathematics, the Milnor conjecture was a proposal by John Milnor (1970) of a description of the Milnor K-theory (mod 2) of a general field F with characteristic different from 2, by means of the Galois cohomology of F with coefficients in Z/2Z. It was proved by Vladimir Voevodsky.
In mathematics, and more specifically in abstract algebra, a rng is an algebraic structure satisfying the same properties as a ring, but without assuming the existence of a multiplicative identity. The term rng is meant to suggest that it is a ring without i, that is, without the requirement for an identity element.
In mathematics, a Witt group of a field, named after Ernst Witt, is an abelian group whose elements are represented by symmetric bilinear forms over the field.
GF(2) is the finite field with two elements. Notations Z2 and may be encountered although they can be confused with the notation of 2-adic integers.
Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Fermat's Last Theorem and the modularity theorem were believed to be impossible to prove using current knowledge by almost all contemporary mathematicians.
The Beilinson–Bernstein localization theorem is a foundational result of geometric representation theory, a part of mathematics studying the representation theory of e.g. Lie algebras using geometry.
